Fast Multiscale Diffusion on Graphs

TL;DR

This paper introduces an improved method for multiscale graph diffusion that reduces computational costs by tightening approximation bounds and optimizing polynomial approximations of the exponential of the Laplacian.

Contribution

It provides a tighter error bound for Chebyshev polynomial approximations and a factorization technique to efficiently compute multiscale graph diffusion.

Findings

Reduced computational cost for multiscale diffusion

Tighter bounds improve approximation accuracy

Enhanced efficiency in graph signal processing

Abstract

Diffusing a graph signal at multiple scales requires computing the action of the exponential of several multiples of the Laplacian matrix. We tighten a bound on the approximation error of truncated Chebyshev polynomial approximations of the exponential, hence significantly improving a priori estimates of the polynomial order for a prescribed error. We further exploit properties of these approximations to factorize the computation of the action of the diffusion operator over multiple scales, thus reducing drastically its computational cost.